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Quotienting by an intermediate ideal is the same as quotienting in one step.
Third isomorphism theorem (rings): Let be a ring and let be ideals of . Then is an ideal of , and there is a natural ring isomorphism
of quotient rings.
This formalizes the idea that “modding out in stages” is equivalent to modding out by the total relation.
An ideal in a ring is an additive subgroup such that it is stable under multiplication by elements of on one side:
A ring isomorphism is a ring homomorphism that is bijective and whose inverse function is also a ring homomorphism.
Let be a ring and let be a two-sided ideal of . The quotient ring is the quotient set of by the equivalence relation iff , with addition and multiplication on cosets defined by
The two-sided condition ensures multiplication is well-defined.
The canonical projection is a surjective ring homomorphism with kernel , and satisfies the universal property of quotients.